Remark 6.4 
To have that \(\nabla F\)  is a multiplicative vector field it is very important to us since this is equivalent to require that the pair of flows \((\Phi ^{\nabla (s^\ast f)}_{\tau },\Phi ^{\nabla f}_{\tau })\)  induces (local) automorphisms on the Lie groupoid; see [1]}. Namely, the following identities hold
\(s\circ \Phi ^{\nabla (s^\ast f)}_{\tau }=\Phi ^{\nabla f}_{\tau }\circ s,\qquad t\circ \Phi ^{\nabla (t^\ast f)}_{\tau }=\Phi ^{\nabla f}_{\tau }\circ t, \qquad \Phi ^{\nabla (f\circ s)}_{\tau }\circ m=m\circ (\Phi ^{\nabla (f\circ s)}_{\tau }\times \Phi ^{\nabla (f\circ s)}_{\tau }).\) 

